Higher educational attainment (EA) is known to have a protective effect regarding the severity of schizophrenia (SZ). However, recent studies have found a small positive genetic correlation between EA and SZ. Here, we investigate possible causes of this counterintuitive finding using genome-wide association results for EA and SZ (n = 443,581) and a replication cohort (1,169 controls and 1,067 cases) with high-quality SZ phenotypes. We find strong genetic overlap between EA and SZ that cannot be explained by chance, linkage disequilibrium, or assortative mating. Instead, our results suggest that the current clinical diagnosis of SZ comprises at least two disease subtypes with non-identical symptoms and genetic architectures: One part resembles bipolar disorder (BIP) and high intelligence, while the other part is a cognitive disorder that is independent of BIP.